66 LECTURES AND ESSAYS [1869-1873] 



the great circle perpendicular to the line joining the 

 sources. 



For the stream lines we have in this case, 



observing that 



az 



, - az 

 tan d&amp;gt; = 



-5- j- 

 xk + hy 



This equation becomes 



a cone which intersects the tangent plane to the sphere 

 at the extremity of the axis of x, in a series of similar 

 ellipses, having their centres on the intersection of the 

 plane with the plane of xz, and passing through the 



points a, ^, 0. Two of the stream lines are manifestly 



great circles, whose equations are x = and z = 0. 



If we divide the sphere along the former of these circles, 

 we cut off the subsidiary source and sink, and get the case 

 of a hemisphere, in which the source and sink are equi 

 distant from the pole. A curious hemispherical case is 

 got by dividing the sphere along the equipotential hemi 

 sphere. In this case we have two sources of the same 

 sign within the hemisphere, one being the subsidiary 

 source of the removed sink. But in order that the 

 distribution may remain unchanged, we must have the 

 potential maintained constant at the edge of the hemi 

 sphere. This may be effected by making the base a 

 conductor with a sink at its centre, or, indeed, by placing 

 the sink at the vertex of any conducting surface of revolu 

 tion which joins the hemisphere. From these hemisphere 

 cases, obvious cases of half and quarter hemispheres 

 follow. 



